Concept

Symbolic dynamics

The description of an orbit by the sequence of regions it visits, turning a map into a shift on strings of symbols. It converts questions about orbits into questions about words, and for a map that doubles it makes the orbit literally the digits of the starting point.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The orbit of 13/32 under doubling, and its word. A cobweb of the doubling map with one orbit drawn, the interval split in half beneath it, and the letter each step contributes written out in order.

The orbit written as a word

Cut the interval in two and record which half each step of an orbit lands in. The orbit becomes an infinite string of two letters, the map becomes the act of deleting the first letter, and questions about trajectories turn into questions about words.

dynamics · symbolic dynamics

The tent map and the logistic map, joined by a change of coordinate. Two cobweb diagrams side by side — the tent map at slope two and the logistic map at four — with the orbit of one carried to the orbit of the other by a curve drawn between them.

The same map in different coordinates

The tent map and the logistic map at four look nothing alike and are the same map, carried onto each other by a change of variable. Everything either one does the other does, and the change of variable is a sine squared.

dynamics · iteration

Named alongside it

The objects these essays reach for when they reach for this one.

Conjugacy Periodic orbit Binary expansion Change of variable Dense orbit Doubling map Homeomorphism Iteration Itinerary Logistic map Lyapunov exponent Shift map

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